# Fixed-income Securities Lecture 2: Basic Terminology and Concepts. Present value (fixed interest rate) Present value (fixed interest rate): the arb

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2 Deep theoretial insights Buy if it s too heap. Sell if it s too expensive. Use an arb to onvert a good deal into pure profit. Spot rate and forward rates So far, we have taken the yield urve to be flat; that is, we have assumed that we an enter ommitments now to borrow and lend money a year from now on the same terms that are available now. That is why the aumulation of interest on a two-year loan is the same in both years. However, this was just a simplifying assumption. In fat, ommitments for a year from now will be typially be made at a different rate than is available now in the spot market. The spot rate is the rate at whih we borrow and lend now for payment a year from now. The rates at whih we ommit for borrowing and lending in different future years are forward rates. We need to fix notation to quantify these things. We will denote by r t the spot rate quoted at t 1 for lending until t. And, we will denote by f(s, t) the forward rate quoted at time s for lending from t 1 to t in the future. It is useful to think of the spot rate as a trivial ase of forward rate, that is, r t = f(t 1, t). A plot of an interest rate against maturity is alled a yield urve. One yield urve is the yield urve in forward rates. We will enounter other yield urves later. PV with a nontrivial yield urve One general expression for the present value with a nontrivial yield urve has disounting by eah of the one-period forward rates in turn. If we have a riskless ash flow that will pay 1 one year hene, 2 two years hene, 3 three years hene, et., then the present value is P V = f(0, 1) + 2 (1 + f(0, 1))(1 + f(0, 2)) = T t= (1 + f(0, 1))(1 + f(0, 2))...(1 + f(0, T )) t t s=1 (1 + f(0, s)) T NPV with a nontrivial yield urve The NPV inludes the initial ash flow. If we have that p is the purhase prie so that 0 = p is the initial ash flow, then the net present value is, as usual, NP V = P V p = T t=0 where an empty produt is defined to equal 1. t t s=1 (1 + f(0, s)) where indiates a sum of terms and indiates a produt.

3 PV: the arb Suppose ash flows of 1 a year hene and 2 two years hene are selling for the bundled prie of p, and this prie is larger than the present value omputed using spot and forward rates f(0, 1) = r 1 and f(0, 2) = r 2. Then we an sell the ash flows in the market (beause they are too expensive) and onvert this profitable opportunity into ash up front using the following arb. sell the ash flow p 1 2 spot lending 1 1+r 1 2 (1+r 1 )(1+r 2 ) r 2 0 forward lending r 2 2 total net ash flow p 1 1+r 1 2 (1+r 1 )(1+r 2 ) 0 0 In-lass exerise: the PV arb Suppose the market prie is 150 for a seurity paying 100 a year from now and 90 two years from now. Further suppose that the spot one-year rate is 25% and the forward rate for lending from one year out to two years out is 20%. Compute the PV and NPV of the ash flow. How do we profit from the disrepany between the market prie and PV? Construt an arb to onvert the profit into a sure thing. sell the ash flow The bad news At this point, it may seem that we have arrived at a very heerful state of affairs: after a few simple alulations we buy and sell and poket ash. Unfortunately, things are not so easy. Absene of arbitrage is the norm. Most arbs still involve some risk-taking. Many smart people are looking for arbs; finding simple arbs is espeially rare. For finane sholars, our theory usually assumes there is no arbitrage, and p = P V. When our theory is wrong, we do not feel so bad, beause then we an make money by trading. Present value using zero-oupon bonds add up pries times quantities as at the groery disount fator D(s, t): prie at date s of getting \$1 at some subsequent date t Look at the orresponding zero-oupon (or bullet or pure-disount) bond with fae F and prie P D(s,t) = P/F Denote by t the ash flow at t. Then, the present value and net present value are given by the following formulas. P V = T t=1 D(0, t) t NP V = T t=0 D(0, t) t

4 Present value using zero-oupon bonds: the arb Consider the following senario. A marketed laim is a riskless self-amortizing loan that pays \$100 a year from now and \$100 two years from now and is pried at \$165 in the market. A zero-oupon bond maturing one year from now osts 90 ents per dollar of fae value, while a zero-oupon bond maturing two years from now osts 80 ents per dollar of fae value. This means that the self-amortizing loan has a present value of.9 \$ \$100 = \$170. Sine the market prie is less than the PV, this laim is heap and we should buy it. Here are the ash flows: buy the ash flow borrowing until year borrowing until year total net ash flow Again, the arb onverts a good deal into pure profit. Zero-oupon pries and yields For any bond, its yield is the fixed interest rate that would make the bond prie equal to the present value of its ash flows. Therefore, the yield z(s,t) at time s on a zero (a zero-oupon bond) maturing at a later date t solves the equation D(s, t) = 1 (1 + z(s, t)) t s where D(s, t) is the orresponding disount fator, so that z(s, t) = D(s, t) 1/(t s) 1 1 D(s, t) t s The approximation is best for short maturities (small t s) and small rates. Institutional note The approximation is similar to, but different from, the formula P = F 1 d dtm 360 used in pratie to relate a Treasury Bill disount d to its prie P, fae value F and days-to-maturity dtm. This formula is not an approximation, and instead it defines what prie orresponds to the quoted yield. Note that 360 is not an approximation to the number of days in the year; Treasury disounts are defined in terms of an artifiial 360-day year. In-lass exerise: disount fators Given the spot one-year interest rate of 20% and a one-year forward interest rate of 25%, what are the one-year and two-year disount fators? What are the one-year and two-year zero-oupon yields?

5 Implied forward rates In the market, we see ative quotes of disount bonds (Treasury STRIPs) rather than the forward borrowing and lending rates. However, we an repliate forward borrowing to obtain an implied forward rate from the STRIP urve. For example, if we want to borrow \$2 million for one year, two years out, we buy \$2 million fae of STRIPs maturing in two years, and we sell short an equivalent dollar value (today) of STRIPs maturing in three years. Given D(0, 2) and D(0, 3), we would buy \$2 million fae of 2-year strips worth \$2D(0, 2) today, whih would be paid for using \$2D(0, 2)/D(0, 3) fae of 3-year strips. The implied forward rate would therefore be (2D(0, 2)/D(0, 3) 2)/2 = D(0, 2)/D(0, 3) 1. The general formula is f(s, t) = D(s, t 1) D(s, t) 1 Implied forward rates: the arb Of ourse, whenever there is a repliating portfolio there is an impliit arb if pries are out of line. For example, suppose we an borrow and lend forward one year from now at 10%, and that the disount fator is 90% one year out and 80% two years out. Then the implied forward rate is.9/.8 1 =.125, whih means (borrowing) money is too heap, so we should buy. At the sale of \$1, the arb is: borrow forward sell one-year zero buy two-year zero total net ash flow Coupon bond yields The yield on a oupon bond is the interest rate that makes its prie equal to its present value. If a bond has prie P, T remaining oupons of per period, and fae value F (paid at the maturity T periods from now), then the yield y solves the equation In-lass exerise: par oupon yields A oupon bond maturing in two years has a fae value of \$112 million and an annual oupon of \$8 million per year. If its yield is 20%, what is its prie now? (The oupon this year has already been paid, so the prie does not inlude a laim to the oupon now.) P = F (1 + y) + T T s=1 (1 + y) s We often talk of the moneyness of a bond. A bond sells at a disount if its prie is less than its fae value, at par if its prie is equal to its fae value, or at a premium if its prie is greater than its fae value. Institutional note In pratie, most oupon bonds pay oupons twie a year. For this reason, bond yields are ustomarily omputed using half-year rates with interpolation for frations of half-years. The rates are quoted as annual rates, defined to be twie the half-year rates. Also, bonds are traded with arued interest so the prie does not jump at oupon dates.

6 Par oupon bond yields Often when we hear about the yield urve, the speaker has in mind a plot of oupon bond yields against maturity. However, there is no reason in general why different oupon bonds with the same maturity would have exatly the same yield; in fat this would be the exeption rather than the rule. Therefore, when people talk about the yield urve for oupon bonds, they have not really pinned it down. Talking about the par oupon bond yield pins it down. To ompute a par oupon yield from disount fators, we first note that the yield y is the same as the oupon yield /F for a par oupon bond, that is, F = F (1 + y) + T T s=1 (1 + y) s implies y = /F (as an be proven using the annuity formula). Now, so that y = 1 D(0, T ) T s=1 D(0, s) F = F D(0, T ) + T yf D(0, s), s=1

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